Question about a charge injection compensation network for analog switches
I think the complete circuit for the image you show should look like this.







simulate this circuit &ndash; Schematic created using CircuitLab

The circuit works in two phases, \$\phi\$ and \$\overline{\phi}\$. The figure indicates the phase when the switches are on.
In the \$\overline{\phi}\$ Phase
The input voltage is sampled on the capacitor \$C_2\$. Note that \$V_{out}\$ will be ground due to the feedback. So, you cannot use an operational amplifier for its implementation but a \$G_m\$ stage.
In the \$\phi\$ Phase
Capacitor \$C_2\$ is shorted and all its charge appears on the \$C_1\$ capacitor, giving a gain of \$\frac{C_2}{C_1}\$.
This type of circuit is useful since it combines sampling and amplification of input voltage in one circuit which relaxes the resolution requirement of the following stage in ADC.
I think you already understand how it cancels the charge injection. So, hopefully it answers your question.