Hom-Category

Let $\mathcal{C}$ be a 2-category, and let $a,b\in \mathcal{C}$. The hom-cagegory $\mathcal{C}[a,b]$ is the category where:

• objects are $\mathcal{C}$-morphisms $f:a\to b$.
• morphisms are 2-morphisms $F:f\to g$ for $f,g:a\to b$