Is the composition of hom-spaces entirely determined by the tensor in a simplicial model category?

Question

Let $\mathcal{C}$ be a simplicial model category, as defined in https://ncatlab.org/nlab/show/simplicial+model+category. We can prove that the $\mathrm{Hom}$ functor is determined by the tensor, because we have that $\mathrm{Hom}(A,B)$ is naturally isomorphic to the simplicial set $\mathcal{C}(A \otimes \Delta[-], B)$. But can we also describe the composition with the tensor? Notably, is it true that the composition of two morphisms $f : A \otimes \Delta[n] \to B$ and $ g : B \otimes \Delta[n] \to C $ can be given at level $n$ by the composition of the maps $$ A \otimes \Delta[n] \to A \otimes (\Delta[n] \times \Delta[n]) \to (A \otimes \Delta[n]) \otimes \Delta[n] \to B \otimes \Delta[n] \to C $$ where the first map is given by the diagonal map $\Delta[n] \to \Delta[n] \times \Delta[n]$, as it is in the simplicial structure on $\mathrm{Top}$? And if not, can we at least prove that the tensor determines the composition up to natural isomorphism?