Let $ X $ and $ Y $ be differentiable manifolds, and let $ f\colon X\to Y $ be a smooth map.
Given $ x\in X $ one can define the canonical pullback at $ x $ map $$ f_x^*\colon \mathscr C_{Y,f(x)}^\infty\to \mathscr C_{X,x}^\infty $$ between the stalks of the sheaf $ \mathscr C_Y^\infty $ of smooth functions on $ Y $ based at $ f(x) $, and the stalk of smooth functions on $ X $ based at $ x $. Recall that $ f_x^* $ maps a germ $ t_{f(x)} $ of representative $ (V,t) $ to the germ represented by $ (f^{-1}(V),t\circ f) $.
The map $ f_x^* $ has the following functorial-like properties:
- $ (1_X)_x^* = 1_{\mathscr C_{X,x}^\infty} $;
- $ (g\circ f)_x^* = f_x^*\circ g_{f(x)}^* $;
In particular this means that there exists a functor $$ F\colon \mathrm{Diff(*)}^{\mathrm{op}}\to \mathrm{Alg}_{\mathbb R} $$ from the category of pointed manifolds to the category of algebras over the real numbers mapping $ (X,x) $ to the stalk $ \mathscr C_{X,x}^\infty $ and a smooth map $ f\colon (X,x)\to (Y,y) $ to $ f_x^*\colon \mathscr C_{Y,y}^\infty\to \mathscr C_{X,x}^\infty $.
Now, properties 1. and 2. can be easily checked by hand. Nevertheless am looking if there is a way to realize $ F $ as a composition of "known" functors.
P.S. I'm familiar with sheaf theory and with the theory of (locally) ringed spaces. $ F $ can be defined for those kind of spaces also, and I would be really happy to see an answer at that level of generality.