Let $P_1, P_2$ be two $k$-algebras and denote by $\Delta_{P_i}\colon P_i\times P_i \to P_i$ $(i=1,2)$ their multiplications. Let $P_1 \otimes_k P_2$ be the tensor product of $P_1$ and $P_2$. Is the multiplication in $P_1 \otimes_k P_2$ the tensor product, in any reasonable sense, of $\Delta_{P_1}$ and $\Delta_{P_2}$?
My worry comes from the fact that I am not used to tensoring maps other than algebra homomorphisms.