I am thinking about the universal property of products:

Let $X$ and $Y$ be objects of a category $D$. The product of $X$ and $Y$ is an object $X \times Y$ together with two morphisms

$\pi\_1 : X \times Y \rightarrow X$

$\pi\_2: X \times Y \rightarrow Y$

such that for any other object $Z$ of $D$ and morphisms $f : Z \rightarrow X$ and $g : Z \rightarrow Y$ there exists a unique morphism $h: Z \rightarrow X \times Y$ such that $f = \pi\_1 \circ h$ and $g = \pi\_2 \circ h$.

Does it mean that of all morphisms, only one satisfies this, but there could be a map which is not a morphims which makes this work?

That is, if any map $h$ makes this diagram commute then it must necessarily be a morphism?

Sorry, if this is obvious - still getting my head around making sense of caterory theoretic formulations.

To explain where I am coming from: my projections and map $f,g$ are morphisms in category I am working in (say $A$, and I know finite products exists in $A$), I have found a $h$ in conceivably a different category (say $B$) which makes everything commute. I am trying to see if it's possible to argue by virtue of everything else being in $A$, that $h$ must also be a morphism in $A$. There's probably a silly counter example to this that I am missing.