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?
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