Why is $(\boldsymbol{P}\boldsymbol{\lambda}\boldsymbol{P}'+\boldsymbol{D})=\boldsymbol{P}(\boldsymbol{\lambda}+\boldsymbol{D})\boldsymbol{P}'$ True

Question

Let the following matrix have an eigendecomposition, $\boldsymbol{A} = \boldsymbol{P}\boldsymbol{\lambda}\boldsymbol{P}'$ and let $\boldsymbol{D}=d\boldsymbol{I}$ where $\boldsymbol{I}$ is the identity matrix and $d$ is a scalar.

Then, it is true that $(\boldsymbol{A}+\boldsymbol{D})=(\boldsymbol{P}\boldsymbol{\lambda}\boldsymbol{P}'+\boldsymbol{D})=\boldsymbol{P}(\boldsymbol{\lambda}+\boldsymbol{D})\boldsymbol{P}'$, but only under the special condition that the diagonal elements of $\boldsymbol{D}$ are constant.

While true, I'm not certain what formal property allows me to move $\boldsymbol{D}$ alongside the eigenvalues. Thank you for any clarifications.