I am trying to find whether the polynomial (monomial) functor $P : X \rightarrow X\times X $, i.e. $P(X) = X^2$, is faithful, in other words, that if there exists an isomorphism $A\times A \overset {i} {\hookrightarrow} B \times B$, then there is also an isomorphism $A \overset {j} {\hookrightarrow} B$.

I understand that it is important to use the fact that those are squares, i.e. products of isomorphic objects, else it doesn't hold (indeed, $A \times A = (A\times A)\times 1$, etc.).

However I have trouble to figure out the path of the proof. I'm not even sure the property holds outside of the category $Sets$.

I am trying to find whether the polynomial (monomial) functor $P : X \rightarrow X\times X $, i.e. $P(X) = X^2$, is monomorphic on objects, in other words, that if there exists an isomorphism $A\times A \overset {i} {\hookrightarrow} B \times B$, then there is also an isomorphism $A \overset {j} {\hookrightarrow} B$.

I understand that it is important to use the fact that those are squares, i.e. products of isomorphic objects, else it doesn't hold (indeed, $A \times A = (A\times A)\times 1$, etc.).

However I have trouble to figure out the path of the proof. I'm not even sure the property holds outside of the category $Sets$.