What does descent mean? I was telling my linear algebra student the following example: suppose you have a linear map $f: V \to W$, and there is a subspace $V' \In V$ such that $f|_{V'}=0$, then $f$ descent to $V/V'$. In general, it means you define something on a 'cover' of an object, and you want to obtain the thing on the object itself.

Zariski Descent A slightly non-trivial example is the Zariski descent for coherent sheaves. Suppose $X$ is a scheme, and $X = \cup_{i=1}^N U_i$ by some finite open cover $j_i: U_i \to X$, then the restriction $$ L = \prod_i j_i^*: QCoh(X) \to \prod_i QCoh(U_i) $$ which is a left-adjoint (hence called L), and its right-adjoint $R = \oplus_i (j_i)_*$ (here I use the finiteness condition to equate $\prod$ and $\oplus$). Then, we have a comonad $$ \Omega := LR, \quad \Omega \to id, \quad \Omega \to \Omega \circ \Omega. $$