/sci/ - Science & Math

Consider the products [math]X\times Y[/math] of two sets. If two elements [math]a,b\in X\times Y[/math] have the same image under both projections then they are the same. I want a nice way to prove this.
Here's what I have, consider a and b as maps [math]\{1\}\rightarrow X\times Y[/math], then if their composites with the projections are the same, the maps must also be the same. This is fine. My only problem is that I want this to work in say the category of groups as well. How can I adjust this argument to work in "sets with some more structure" categories?