Universal property of algebraic differentials

Every $g\in {\mathrm{Hom}}_B({\mathrm{\Omega }}_{B\mathrm{/}A},M)$ gives $g\circ \mathrm{d}\in {\mathrm{Der}}_A(B,M)$: additivity, $A$-constancy and the Leibniz rule for $g\circ \mathrm{d}$ are those of $\mathrm{d}$ transported by the additive map $g$, and $\mathrm{d}$ has them by construction. The assignment is additive and $B$-linear: $(g_1+g_2)\circ \mathrm{d}=g_1\circ \mathrm{d}+g_2\circ \mathrm{d}$ and $(bg)\circ \mathrm{d}=b(g\circ \mathrm{d})$ pointwise. It is injective, because the classes $\mathrm{d}b$ generate ${\mathrm{\Omega }}_{B\mathrm{/}A}=F\mathrm{/}R$ as a $B$-module, so $g\circ \mathrm{d}=0$ forces $g=0$.

Conversely let $D\in {\mathrm{Der}}_A(B,M)$. Since $F$ is free on the $[b]$, there is a unique $B$-linear $\tilde{D}\text{\hspace{0.17em}⁣}:F\to M$ with $\tilde{D}([b])=D(b)$. Each relator is killed: $\tilde{D}([b+b^{\mathrm{\prime }}]-[b]-[b^{\mathrm{\prime }}])=D(b+b^{\mathrm{\prime }})-D(b)-D(b^{\mathrm{\prime }})=0$ by additivity of $D$, similarly the Leibniz relator, and $[\phi (a)]\mapsto D(\phi (a))=0$ by $A$-constancy. Hence $\tilde{D}$ factors through $F\mathrm{/}R={\mathrm{\Omega }}_{B\mathrm{/}A}$, giving a $B$-linear $g_D\text{\hspace{0.17em}⁣}:{\mathrm{\Omega }}_{B\mathrm{/}A}\to M$ with $g_D(\mathrm{d}b)=D(b)$, that is, $g_D\circ \mathrm{d}=D$.

The two assignments are mutually inverse: the derivation attached to $g_D$ is $g_D\circ \mathrm{d}=D$, and the homomorphism attached to $g\circ \mathrm{d}$ agrees with $g$ on every generator $\mathrm{d}b$, hence equals $g$ because those generators span. Each assignment is additive, so the bijection is a $B$-module isomorphism; and for $B$-linear $h\text{\hspace{0.17em}⁣}:M\to N$ one has $h\circ (g\circ \mathrm{d})=(h\circ g)\circ \mathrm{d}$, which is exactly naturality in $M$.