Finding the construction of universal properties

Question

Identify for all of the following universal properties the corresponding construction (commutative diagram) . Notation: $K$ is a field, $ U $ and $ V $ are vector spaces and $ f:V\rightarrow U$ is a linear map

a) Let $ W $ be a $ K$ vector space with linear map $g:V\rightarrow W$, such that $g\circ f=0$ and there exists for all $K$ vector spaces $T$ with linear mapping $ h:V \rightarrow T$ and $h\circ f = 0$ a unique linear map $l:W \rightarrow T$ with $l \circ g = h.$

So this is one of many exercices our linear algebra teacher gave us to solve. I really do not understand how to attempt an exercice like this. I have very little experience with category theory but maybe you can help me with this one to solve the other ones.

Answer 1

The full exercise could be, with $f:U\to V$ and $g:V\to W$ with the mentioned property:

Prove that $W\cong \def\coker{V/{\,\rm im\,}f} \coker$.

For this, the easiest way is to

1. Prove that $\coker$ satisfies the given universal property.
2. Use the uniqueness criterium to prove that if $W_1$ and $W_2$ both satisfy the universal property, then $W_1\cong W_2$.