Why a dotted line in functions?

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My question is a little stupid, but I have to ask (I've never seen in anywhere an explanation) I'm studying a very short introduction of category theory by Lee's book and I would like to ask why some authors use dotted lines like this one below, maybe because of the unicity of $f$?

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1 Answer

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In general, in a commutative diagram, the solid arrows are the given ones, and the dotted arrows are those which are claimed to exist. You can imagine this as a process:

First there is no arrow: $~X ~~~~~~~~ Y $

The arrow emerges: $~~~~~X \cdots{\small >} Y$

And finally it's there: $~~~~X \longrightarrow Y$

The unicity is not assumed. See for example the definition of a projective module.

When one wants to indicate unicity, one writes for example $X \stackrel{\exists !}{\cdots {\small >}} Y$. For example, here is the fundamental theorem on homomorphisms:

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