He entered the room [smiling].
 I saw him smiling.
In  "smiling" is a depictive adjunct. It's predicative because it refers to a predicand, i.e. "he", and it's an adjunct because it's integrated into clause stucture.
In  "see" is a catenative verb and "smiling" is a subordinate non-finite clause functioning as complement of "saw". Syntactically, the intervening NP "him" is direct object of "saw", and the semantic (understood) subject of the subordinate clause. "Him" is thus a raised object because the verb that it relates to syntactically is higher in the constituent structure than the one it relates to semantically.
Note that "smiling" is not an objective complement. Predicative complements are generally restricted to NPs and AdjPs, but "smiling" is obviously not a noun and it fails the usual tests for adjectivehood, so it must be a verb phrase.
The word 'catenative' comes from the Latin word for "chain", which is appropriate here since the verbs "saw" and "smiling" do form a chain, separated only by the NP "him".
NOTE: "Smiling" does not qualify as an adjective. We can compare it to the genuine participial adjective "entertaining".
"Smiling" can't be modified by "very". We can't say *a very smiling child, but we can say a very entertaining clown.
"Smiling" can't occur as complement to complex-intransitive verbs like "become" or "seem". We can't say *He became/seemed quite smiling, but we can say He became/seemed quite entertaining:
"Smiling" can't occur as complement to complex-transitive verbs like "find". We can't say *I found him quite smiling, but we can say I found him quite entertaining.
The range of expressions that can occur as pre-head modifier to a noun is very large and varied: we don't want to call them all adjectives. The same applies to other uses of verbs like "smiling", where it is equally clear that it can't be an adjective.
"Entertaining" has the properties of indisputable adjectives and hence must belong in that class, but "smiling" doesn't have those properties and hence is a verb phrase in .