/sci/ - Science & Math

What is it that makes category theory autistic?
How come abstract, axiomatic systems feel less autistic than fucking diagrams?

Quick question. How to define unique morphism from pushout in category of sets? Picrelated's a picture from Wiki and I'll proceed with using naming from it.

So, [math]P[/math] is defined as a disjoint union of [math]X[/math] and [math]Y[/math] with [math]x \in X'[/math] being equivalent to [math]y \in Y'[/math] if [math]x = f(z)[/math] and [math]y = g(z)[/math] for some [math]z \in Z[/math] where [math]X', Y'[/math] are isomorphic copies in disjoint union. Now, I have a question on how to define the unique morphism [math]u: P \rightarrow Q[/math]. Can I just say that [math]u([p]) = j_1([p])[/math] if the representative of an equivalence class happens to be from [math]X'[/math] and [math]u([p]) = j_2([p])[/math] otherwise? I don't know how properly define [math]u[/math] so that the choice of representative of an equivalence class doesn't matter. Moreover, should I provide a concrete construction of a disjoint union (like [math]X \times \{0\} \bigcup X \times \{1\}[/math])? Because if I don't do this, I technically cannot define [math]u[/math] by using [math]f, g[/math] because I work with isomorphic copies and cannot extract element from original sets.

Anytime I do maths and try to prove a theorem, I start biting my nails nervously. It's pretty much the only occasion I bite them and it makes them look really bad. How do I stop?

I don't understand the difference between creating and reflecting limits in category theory as defined on p. 151 in Peter Smith's introductory textbook. Theorems 95 and 96 seem to confirm this but apparently creating is a stronger condition than reflecting. Please help me out