Suppose $f:X\rightarrow Y$ is a morphism of schemes; is there a computationally cleaner way of working with $f^{-1}\mathcal{O}_Y$ than just using the raw definition? Anytime I need to use the inverse image functor I end up in a notational quagmire because I'm dealing with the sheafification of a presheaf defined in terms of a colimit, so I've got two colimits to work, over different indexing sets, and it's just a mess.
In particular, there are certain cases where, with some work with the raw definition, you can show you don't completely need it. For example, if $Z\subset X$ is a Zariski closed subset, then we get a sheaf of ideals $I_Z$ of $X$ determined by $Z$, and which induces a new sheaf of rings on $X$, denoted $\mathcal O_X/I_Z$, which for affine opens satisfies $\mathcal O_X/I_Z(U)\cong \mathcal O_X(U)/I_Z(U)$. The sheaf $\mathcal O_Z=\iota_{Z}^{-1}\mathcal O_X/I_Z$ on $Z$ is the one which gives $Z$ its induced reduced subscheme structure, and is a pain to work with directly. But, for every open affine of $U$ it satisfies $\mathcal{O}_Z(U\cap Z)\cong \mathcal O_X(U)/I_Z(U)$, so we never need to really work with the inverse image sheaf directly.
I doubt something as clean would hold in generality, but is there any argument I can make in specific cases that allows me to avoid working with the definition directly? In particular, I'd be very interested in the case of the definition of the pull back of a sheaf $G$ of $\mathcal{O}_Y$ modules: $$f^*G=f^{-1}G\otimes_{f^{-1}\mathcal{O}_Y}\mathcal O_X$$ which feels incredibly unwieldy to work with from the raw definitions.
I'd also be interested in the specific sub case of the sheaf of differentials on $X$ over $Y$, which is defined by: $$\Omega_{X/Y}=\iota^*(I_{X/U}/I_{X/U}^2)$$ where $\iota:X\rightarrow U$ is the closed embedding in the diagram: $$\Delta_{X/Y}:X\hookrightarrow U\hookrightarrow X\times_YX$$ and $I_{X/U}$ is the sheaf of ideals determined by the image of $\iota$.
Any advice or references would be greatly appreciated.