One very useful example is the following: assume $\mathcal{T} = \mathrm{D}^b(\mathrm{coh}(X))$, and $\mathcal{T}' = \mathrm{D}^b(\mathrm{coh}(X'))$, where $X$ and $X'$ are smooth projective varieties. Then $\mathcal{T}' \subset \mathcal{T}$ is admissible, because in the geometric case the fully faithful functor $$ \mathrm{D}^b(\mathrm{coh}(X')) \hookrightarrow \mathrm{D}^b(\mathrm{coh}(X)) $$ is a Fourier--Mukai functor by Orlov's representability theorem, and any Fourier--Mukai functor between smooth projective varieties has both left and right adjoints.

One very useful example is the following: assume $\mathcal{T} = \mathrm{D}^b(\mathrm{coh}(X))$, and $\mathcal{T}' = \mathrm{D}^b(\mathrm{coh}(X'))$, where $X$ and $X'$ are smooth projective varieties. Then any fully faithful embedding $\mathcal{T}' \hookrightarrow \mathcal{T}$ is admissible, because in the geometric case any fully faithful functor $$ \mathrm{D}^b(\mathrm{coh}(X')) \hookrightarrow \mathrm{D}^b(\mathrm{coh}(X)) $$ is a Fourier--Mukai functor by Orlov's representability theorem, and any Fourier--Mukai functor between smooth projective varieties has both left and right adjoints.