In $\left[\mathrm{Co}\right({\mathrm{NH}}_3{)}_6{]}^{3+}$, $\mathrm{Co}$ is in $+3$ oxidation state with the configuration $3{\mathrm{d}}^6$. In the presence of ${\mathrm{NH}}_3$ a strong field ligand, the 3d electrons undergo pairing and  leaving two d-orbitals empty. Hence, the hybridisation is ${\mathrm{d}}^2{\mathrm{sp}}^3$ which forms an inner orbital octahedral complex. The complex is diamagnetic as it has no unpaired electrons.

So, the correct option is C