symmetric monoidal (∞,1)-category of spectra
A Lawvere theory is encoded in its syntactic category $T$, a category with finite products such that all objects are finite products of a given object.
An algebra over a Lawvere theory $T$, or $T$-algebra for short, is a model for this algebraic theory: it is a product-preserving functor
The category of $T$-algebras is the full subcategory of the functor category on the product-preserving functors
For more discussion, properties and examples see for the moment Lawvere theory.
The category $T Alg$ has all limits and these are computed objectwise, hence the embedding $T Alg \to [T,Set]$ preserves these limits.
$T Alg$ is a reflective subcategory of $[T, Set]$:
With the above this follows using the adjoint functor theorem.
The category $T Alg$ has all colimits.
for more see Lawvere theory for the moment
rings
$k$-associative algebras
algebra over an algebraic theory