Suppose you have $F$ a subbundle of a vector bundle $E\to X$ over a scheme $X$. Recall that there is a scheme $\varphi\colon Y\to X$ with the points of $Y$ over the point $x\colon\mathrm{spec}(A)\to X$ are the morphisms $x^*(E)\to x^*(F)$ which restrict to the identity on $x^*(F)$.
I have two hiccups in my reading: is there an explanation of why $Y$ is affine, with $\varphi^*(F)$ a direct summand of $\varphi^*(E)$?