Monoid objects in a cartesian category are internal monoids; in Set they are ordinary monoids

By [L1], the tensor is $M\times M$ and the unit is a terminal object $1$. So [L2] says exactly that a monoid object on $M$ is data $\mu :M\times M\to M$ and $e:1\to M$ satisfying the usual associative and left and right unit diagrams, with the only extra notation being the canonical rebracketing isomorphism $((M\times M)\times M)\cong M\times (M\times M)$.

In $\mathbf{S}\mathbf{e}\mathbf{t}$, a morphism $1\to M$ is the choice of an element $e\in M$, and a morphism $M\times M\to M$ is a binary operation on the underlying set. The three diagrams from step 1.1 then say exactly $\mu (\mu (x,y),z)=\mu (x,\mu (y,z))$ and $\mu (e,x)=x=\mu (x,e)$ for all $x,y,z\in M$.

Therefore monoid objects in a cartesian monoidal category are associative unital multiplications internal to that cartesian structure, and in $\mathbf{S}\mathbf{e}\mathbf{t}$ they are exactly ordinary monoids.