The functor $M{\otimes }_A-$ is additive, right exact, and preserves direct sums over an arbitrary unital ring

Additivity: for parallel maps $u,v:X\to Y$ and every elementary tensor, $(1\otimes (u+v))(m\otimes x)=m\otimes (u+v)(x)=m\otimes u(x)+m\otimes v(x)=(1\otimes u)(m\otimes x)+(1\otimes v)(m\otimes x)$; both sides are homomorphisms out of $M{\otimes }_{A}X$, so they are equal by [F3]. Hence $T_M$ preserves addition of morphisms and is additive.

Direct sums, first map: by [F5] the maps $1_M\otimes {\mathrm{ȷ}}_i:M{\otimes }_A{X}_i\to M{\otimes }_A({⨁}_j{X}_j)$ induce a unique homomorphism $\mathrm{\Phi }:{⨁}_{i\in I}(M{\otimes }_A{X}_i)\to M{\otimes }_A({⨁}_{i\in I}{X}_i)$ whose composite with the coordinate inclusion ${\mathrm{ȷ}}_i^{\mathrm{\prime }}:M{\otimes }_A{X}_i\to {⨁}_i(M{\otimes }_A{X}_i)$ is $1_M\otimes {\mathrm{ȷ}}_i$ for every $i$.

Direct sums, inverse: the pairing $b(m,(x_i{)}_i):=(m\otimes x_i{)}_i$ is well defined and balanced because the family $(x_i)$ has finite support, addition is coordinatewise, and $b(ma,(x_i))=(ma\otimes x_i{)}_i=(m\otimes (a{x}_i){)}_i=b(m,a(x_i))$; by [F1] it induces $\mathrm{\Psi }:M{\otimes }_A({⨁}_i{X}_i)\to {⨁}_i(M{\otimes }_A{X}_i)$ with $\mathrm{\Psi }(m\otimes (x_i{)}_i)=(m\otimes x_i{)}_i$.

Cokernels, surjectivity and composite: $1\otimes g$ is surjective, since every element of $M{\otimes }_{A}Z$ is a finite sum of elementary tensors $m\otimes z$ and $z=g(y)$ for some $y$ by [F6], so it is the image of $\sum m\otimes y$. Also $(1\otimes g)\circ (1\otimes f)=1\otimes (g\circ f)=1\otimes 0=0$, because $g\circ f=0$ by [F6] and a homomorphism out of $M{\otimes }_{A}X$ vanishing on every elementary tensor is zero by [F3].

Cokernels, universal property: let $W$ be an abelian group and $v:M{\otimes }_{A}Y\to W$ a homomorphism with $v\circ (1\otimes f)=0$. For $z\in Z$ choose $y\in Y$ with $g(y)=z$ and set $c(m,z):=v(m\otimes y)$. If $y^{\mathrm{\prime }}$ is another lift then $y-y^{\mathrm{\prime }}=f(x)$ for some $x\in X$ by [F6] and [F7], so $m\otimes y-m\otimes y^{\mathrm{\prime }}=m\otimes f(x)=(1\otimes f)(m\otimes x)$ by [F2] and [F3], whence $v(m\otimes y)=v(m\otimes y^{\mathrm{\prime }})$: the map $c$ is well defined. It is balanced, being additive in each variable with $c(ma,z)=v(ma\otimes y)=v(m\otimes (ay))=c(m,az)$, so by [F1] it induces a unique homomorphism $w:M{\otimes }_{A}Z\to W$ with $w(m\otimes z)=v(m\otimes y)$; then $w\circ (1\otimes g)=v$, since both sides send $m\otimes y$ to $v(m\otimes y)$, and any $w^{\mathrm{\prime }}$ with $w^{\mathrm{\prime }}\circ (1\otimes g)=v$ satisfies $w^{\mathrm{\prime }}(m\otimes z)=w^{\mathrm{\prime }}(m\otimes g(y))=v(m\otimes y)$, so $w^{\mathrm{\prime }}=w$ by [F3].

Bimodules: if $M$ is a $(B,A)$-bimodule, then $M{\otimes }_{A}X$ is a left $B$-module with $b(m\otimes x)=(bm)\otimes x$ by [F8], and every map considered above is $B$-linear: the induced tensor maps by $(1\otimes f)(b(m\otimes x))=(bm)\otimes f(x)=b(m\otimes f(x))$, and $\mathrm{\Phi },\mathrm{\Psi }$ because their defining pairings and families are $B$-linear in $m$ and $B$-linearity is checked on the generating elementary tensors and coordinate inclusions.

The maps $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are mutually inverse. First, $\mathrm{\Psi }\circ \mathrm{\Phi }$ fixed on the generators ${\mathrm{ȷ}}_i(m\otimes x_i)$ of the direct sum equals ${\mathrm{ȷ}}_i(m\otimes x_i)$, since $\mathrm{\Phi }({\mathrm{ȷ}}_i(m\otimes x_i))=(1\otimes {\mathrm{ȷ}}_i)(m\otimes x_i)=m\otimes {\mathrm{ȷ}}_i(x_i)$ and then $\mathrm{\Psi }(m\otimes {\mathrm{ȷ}}_i(x_i))=(m\otimes x_i{)}_i={\mathrm{ȷ}}_i(m\otimes x_i)$; by [F5] this forces $\mathrm{\Psi }\circ \mathrm{\Phi }=\mathrm{id}$. Second, $\mathrm{\Phi }\circ \mathrm{\Psi }$ and the identity agree on every elementary tensor $m\otimes (x_i{)}_i$, where $\mathrm{\Psi }$ gives the finitely supported family $(m\otimes x_i{)}_i$, $\mathrm{\Phi }$ sends it to ${\sum }_i(1\otimes {\mathrm{ȷ}}_i)(m\otimes x_i)={\sum }_{i}m\otimes {\mathrm{ȷ}}_i(x_i)=m\otimes {\sum }_i{\mathrm{ȷ}}_i(x_i)=m\otimes (x_i{)}_i$, and ${\sum }_i{\mathrm{ȷ}}_i(x_i)=(x_i)$; by [F3] this forces $\mathrm{\Phi }\circ \mathrm{\Psi }=\mathrm{id}$. If $I=\mathrm{\varnothing }$, then ${⨁}_i{X}_i=0$ by [F4], and $M{\otimes }_{A}0=0$: the balanced map $\tau :M\times 0\to M{\otimes }_{A}0$ is zero by [F3], so the identity and the zero endomorphism of $M{\otimes }_{A}0$, which both compose with $\tau$ to $\tau$, are equal by uniqueness in [F1]; the comparison map $0\to M{\otimes }_{A}0$ is then an isomorphism.

Assembling: $T_M$ is additive by step 1.1, preserves cokernels by steps 1.4 and 1.5 (so it carries the given exact sequence to the exact sequence with kernel $\mathrm{im}(1\otimes f)$ and surjective $1\otimes g$, which is right exactness in the stated sequence form), and preserves arbitrary direct sums including the empty one by step 2.2; in the bimodule case step 2.1 shows that $T_M$ takes values in left $B$-modules and that all displayed maps are $B$-linear. No element of an auxiliary family is chosen globally, so no choice is used.