In second semester analysis we learned about the product topology which is quite easy to categorify using limits. However, we also learned about the coinduced topology $\mathfrak{V}$ induced by $f: X → Y$ and $(X, \mathfrak{U})$ on $Y$. It is the strongest topology on Y such that $f: (X, \mathfrak{U}) → (Y, \mathfrak{V})$ is continuous.
I would love to to express this coinduced topology using the categories Top and Set. Sadly I can't realy figure out how to do it as I have few experience with Category Theory.
Could someone please explain this to me?