For complexes $A,B$ of (left) modules over a ring with unit, we let $$\text{Hom}_\text{DG}(A,B) := \bigoplus_i\text{Hom}^i(A,B),\ \text{ where } \text{Hom}^i(A,B):= \bigoplus_v \text{Hom} (A^v, B^{v+i})$$ be the complex with differential $$d: \psi \mapsto d_B \psi - (-1)^{|\psi|}\psi d_A$$ where $\psi \in \text{Hom}^{|\psi|}(A,B)$ is homogenous of degree $|\psi|$.
Let $$\text{Hom}(A,B) := \ker d \cap \text{Hom}^0_\text{DG}(A,B)$$ and $$\text{Hom}_{\mathcal{H}_0}(A,B) := H^0(\text{Hom}_\text{DG}(A,B))=\text{Hom}(A,B)/ \text{im } d$$
I have to prove that complexes with morphisms being the $\text{Hom}$-spaces form an abelian categories, with the morphisms being $\text{Hom}_{\text{DG}}$-complexes form a $\text{DG}$-category, and with the morphisms being the $\text{Hom}_{\mathcal{H}_0}$-spaces form a category.
I'm doing a course where category theory has been just spoken of but hasn't been discussed well, as the course is on sheaf theory. But this is the first time I'm doing category theory or homological algebra. So, I was hoping if someone could explain what exactly is going on here. I understand what is being asked of me i.e. I'm aware of definitions, but not really sure on how to even approach this problem.