Simple problem from category theory

Question

The problem is to prove that for given initial object $X$ and any object $Y$ there is exists canonical isomorphism of $X+Y$ with $Y$, where $X+Y$ means coproduct in category under consideration. It is clear that I just have to draw diagrams (universal property of coproduct), but I can't get the answer from it so far.

Answer 1

The coproduct comes with inclusions $i_1\colon X\to X+Y$ and $i_2\colon Y\to X+Y$.

You also have distinguished morphisms from $X$ and $Y$ to $X$. By universal property of the coproduct they give a morphism $f$ from $X+Y$ to $X$.

Now you should compute the compositions of $i_1$ and $f$.

Answer 2

The only obvious morphisms you can think of are $id\colon Y\to Y$ and the unique morphism $X\to Y$. Now check that they satisfy the universal property of the coproduct $X\sqcup Y$.