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Let $X$ and $Y$ be topological manifolds and let $\{(\phi_x,U_x)\}_{x \in X}$ and $\{(\psi_y,Y_y)\}_{y \in Y}$ be respective atlases of $X$ and $Y$; with each $\phi_x:U_x\rightarrow \mathbb{R}^n,\psi_y:V_y\rightarrow\mathbb{R}^m$ homomorphism onto their images and each $U_x,V_y$ open and non-empty.

Suppose that I'm given $\{f_x\in C(\phi_x(U_x),\mathbb{R}^m)\}_{x \in X}$. Can (when) I find a map $F:X\rightarrow Y$ (just a plane set-function) such that: $$ \psi_{F(x)}\circ f_x\circ \phi_x , $$ is a well-defined element of $C(X,Y)$?

Can this always be done if $X$ and $Y$ are topological manifolds? If not, what if we assume them to be $C^{\infty}$?

Concern: My only real concern is if we can "glue together" the $f_x\circ \phi_x$ by post-composing with the correct $\psi_y$ (as assigned by $F$). I imagine this requires some type of condition on the $\{f_x\}_{x \in X}$?

Note A: I added the tag 'sheaf' since I feel like that may be a reasonable (possible) way to approach this problem.

Note B: The usual condition of requiring that $f_x\circ \phi_x|_{U_x \cap U_y} = f_y\circ \phi_y|_{U_x \cap U_y}$ for all $x$ and $y$ is only necessary in this case?

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    $\begingroup$ I think you've found a complicated way of asking whether continuous functions with values in $Y$ form a sheaf on $X$, viꝫ., whether given continuous functions $U\to Y$ on open sets $U$ covering $X$ which coincide on pairwise intersections, can be pieced together in a unique way to a continuous function $X\to Y$, and the answer is “yes”. All you need to know is this, plus the fact that left or right composing a continuous function by a homeomorphism (your various $φ$ or $ψ$ or their inverses) still gives a continuous function. $\endgroup$
    – Gro-Tsen
    Commented Apr 1, 2021 at 11:57
  • $\begingroup$ Yes; but I want to express the "compatibility condition" in terms of the Euclidean functions $\{f_x\}_x$; which I can't see immediately how to do (without stating some type of tautological condition). (P.s.: ꝫ is cool; I never knew about this scribe thing before). $\endgroup$
    – ABIM
    Commented Apr 1, 2021 at 12:12

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