In a semicartesian monoidal category, any tensor product of objects $x \otimes y$ comes equipped with morphisms $$ p_x : x \otimes y \to x $$ $$ p_y : x \otimes y \to y$$ given by $$ x \otimes y \stackrel{1 \otimes e_y}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x $$ and $$ x \otimes y \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes y \stackrel{\ell_y}{\longrightarrow} y $$ respectively, where $e$ stands for the unique morphism to the terminal object and $r$, $\ell$ are the right and left unitors. We can thus ask whether $p_x$ and $p_y$ make $x \otimes y$ into the product of $x$ and $y$. If so, it is a theorem that $C$ is a cartesian monoidal category. (This theorem is probably in Eilenberg and Kelly's paper on closed categories, but they may not have been the first to note it.) This also follows if we posit the existence of a natural diagonal morphism $x \to x \otimes x$.
Do $p_x$ and $p_y$ not make it into a full-fleged product? What am I missing? Is it that the extra $1 \otimes e_x$/$1 \otimes e_y$ arrows mean that $x \rightarrow p_x (x \otimes x)$ is merely an isomorphism and not the identity?