Proof that equality on categorical products is componentwise equality I want to proof that in the categorical product as defined here it holds that
for $x,y \in \prod X_i$ then
$$
 x = y \textrm{ iff } \forall i \in I : \pi_i(x) = \pi_i(y).
$$
The direction from left to right is trivial, but the other, that iff the components equal than their product is equal I am not able to proof, I tried to substitute the identity morphisms in the universal property, but I always get the wrong "types" in the functions involved. Any hints?

In arbitrary categories, there is a notion of "generalized element": A generalized element of an object $A$ is any morphism into $A$ (from any object of the category). A morphism $A\to B$ can be applied to a generalized element $Z\to A$ just by composing them to get a generalized element $Z\to B$. In these terms, the result you want can be proved: A generalized element of $\prod_iX_i$ is determined by its images under all the projections $\pi_i$. But "proved" here is too grandiose a term; this fact is just part of the categorical definition of product.
