Subobjects in the category of presheaves?

Question

Suppose $\mathcal{C}$ is a locally small category, and $X$ be an element of $\mathcal{C}.$ A sub-object of $X$ is an isomorphism class of monomorphisms in to $X.$ Now suppose we embedd $X$ in $[\mathcal{C}^{op}, \mathbf{Set}]$ using Yoneda $X\mapsto y(X)=\text{Hom}(-,X)$. I would like to understand sub-objects (or sub-functors) of the functor $y(X).$ Without much of luck, I am struggling with this for sometime. Can anyone give me a hint on this problem?

Answer 1

Subobjects of representable functors are the same thing as sieves on the representing objects, i.e. collections of maps into the representing object that are stable under precomposition with maps in $\mathcal C$. The correspondence is established by sending a subfunctor $R\hookrightarrow y(X)$ to the collection $\bigcup_{Y\in \mathcal C} R(Y)$ and conversely by sending a sieve $S$ on $X$ to the functor $R$ that maps an object $Y$ to the set of maps $Y\to X$ that are contained in $S$ and a map $Z\to Y$ to the map $R(Y)\to R(Z)$ given by precomposition.