Suppose $\mathcal{A}$ and $\mathcal{B}$ are categories with products, and $T$ a functor between them. If $X$ and $Y$ are objects in $\mathcal{A}$, what does it mean when we say there is a natural morphism $f\colon T(X\times Y)\to T(X)\times T(Y)$?
In $\mathcal{A}$, we have the product $X\times Y$, with corresponding morphisms $\pi_1:X\times Y\to X$ and $\pi_2:X\times Y\to Y$. Under $T$, we get a diagram of objects in $\mathcal{B}$ of morphisms $T(\pi_1):T(X\times Y)\to T(X)$ and $T(\pi_2):T(X\times Y)\to T(Y)$.
Since products exist in $\mathcal{B}$, we have a product $(T(X)\times T(Y),p_1,p_2)$ such that there is a unique morphism $f\colon T(X\times Y)\to T(X)\times T(Y)$ such that $p_1f=T(\pi_1)$ and $p_2f=T(\pi_2)$.
My guess is that this $f$ is the so called natural morphism, but I don't know how to verify that because I don't know what it means. I've only heard of natural transformations/isomorphisms between functors, but not natural morphisms between objects. Can anyone clarify?