A category $\mathrm{𝒞︀}$ consists of

1. A type of objects ${\mathrm{𝒞︀}}_0$
2. For each pair of objects $𝑥,𝑦:{\mathrm{𝒞︀}}_0$ a set of morphisms $\mathrm{𝒞︀}(𝑥,𝑦)$. We may simply write a morphism with an arrow, denote $𝑓:\mathrm{𝒞︀}(𝑥,𝑦)$ as $𝑓:𝑥\to 𝑦$ or $𝑥\stackrel{𝑓}{\to }𝑦$ or similar
3. A composition operation on morphisms. For $𝑓:𝑥\to 𝑦$ and $𝑔:𝑦\to 𝑧$, there is a morphism $𝑓\star 𝑔:𝑥\to 𝑧$
4. For each $𝑥:{\mathrm{𝒞︀}}_0$, an identity morphism ${\text{𝗂𝖽}}_{𝑥}:𝑥\to 𝑥$

5. Left-unitality of composition: for all $𝑓:𝑥\to 𝑦$, an equality

   $${\text{𝗂𝖽𝖫}}_{𝑓}:{\text{𝗂𝖽}}_{𝑥}\star 𝑓=𝑓$$

6. Right-unitality of composition: for all $𝑓:𝑥\to 𝑦$, an equality

   $${\text{𝗂𝖽𝖱}}_{𝑓}:𝑓\star {\text{𝗂𝖽}}_{𝑦}=𝑓$$

7. Associativity of composition: for all $𝑓:𝑥\to 𝑦$, $𝑔:𝑦\to 𝑧$, $ℎ:𝑧\to 𝑤$, an equality

   $${\text{𝖺𝗌𝗌𝗈𝖼}}_{𝑓,𝑔,ℎ}:(𝑓\star 𝑔)\star ℎ=𝑓\star (𝑔\star ℎ)$$

Concretely, the definition above is meant to model the one used in the Cubical standard library [1].

However, the notion of category is flexible. Depending on the context, we may be talking of small, locally small, wild, or any other kind of category that may augment which things we require to be (homotopy) sets, which things we require to be small types, etc. For the most part, the same idea of a category will apply across all of these settings.