A Note on Operator Equations Describing the Integral

We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type
\begin{equation} f\circ g + c = I\ (Tf\circ g\cdot Tg), \quad f,g\in C^1(\mathbb{R}),\tag{1} \end{equation}
where $T\!: C^1(\mathbb{R})\to C(\mathbb{R})$ and where $I$ is defined on $C(\mathbb{R})$. We consider suitable conditions on $I$ and $T$ such that (1) is well-defined and, after reformulating (1) as
\begin{equation} V(f\circ g)=Tf\circ g\cdot Tg, \quad f,g\in C^1(\mathbb{R})\tag{2} \end{equation}
with $V\!: C^1(\mathbb{R})\to C(\mathbb{R})$, give the general form of $T$, $V$ and $I$. Simple initial conditions then guarantee that the derivative and the integral are the only solutions for $T$ and $I$. We also consider an analogue of the Leibniz rule and study surjectivity properties there.