Just a quick question regarding the notation in the category notes: Which notation are we using,

1)
A set objects: $$𝒜\text{.}$$ The set of morphisms associated to the objects: $${\mathrm{Hom}}_{𝒜}=\left\{{\mathrm{Hom}}_{𝒜}\left(X,Y\right)|X,Y\in 𝒜\right\}\text{.}$$ The category itself: $$\left(𝒜,{\mathrm{Hom}}_{𝒜}\right)\text{.}$$

or

2)
The category itself: $$𝒜\text{.}$$ The set of morphisms associated to the objects/category: $${\mathrm{Hom}}_{𝒜}\text{.}$$ A set objects: $$\text{???}$$

?

The hand written notes seem to mostly follow (1) but to sometimes use (2). If this is intentional abuse, should it be noted?