The first part is my work on a variation of Yoneda Lemma. The second part is my work on **Exercises 2.4A, 2.4.C-2.4.D** of section 2.4 in Vakil’s notes.

**1. Variations of Yoneda Lemmas **(Monos, Epis and Isomorphisms of Presheaves)

Here are a few variations of Yoneda Lemma I played around a few years ago, which bear similar ideas of Yoneda Lemma. Recently I have been dealing with sheaves again, so I just reviewed some old stuff here. I used these variations to show that a morphism of presheaves is monic resp. epic if and only if it’s injective resp. surjective on the level of sections (For another proof see here https://stacks.math.columbia.edu/tag/00V5).

Here are the variations and proofs for presheaves: (pdf version: Variations of Yoneda lemma)

### 2. Monos, Epis and Isomorphisms of Sheaves

Here we give a detailed discussion for sheaves, following exercises in Section 2.4 of Vakil’s notes: (pdf version: Monos, epis and isomorphisms of sheaves)