symmetric monoidal (∞,1)-category of spectra
Given a monad $T \;\colon\; \mathcal{C} \to \mathcal{C}$, its unit is the natural transformation
which is part of the definition of monad. Hence for every object $X \in \mathcal{C}$ the component of the unit on $X$ is a morphism
in $\mathcal{C}$.
Dually, there is a counit of a comonad.
If $(L \dashv R)$ is an adjunction that gives rise to the monad $T$ as $T \simeq R \circ L$, then the unit of the monad is equivalently the unit of the adjunction.
unit of a monad