hope vs be after

hope is a noun but be after is not a noun.

Word	Noun	Adjective	Verb	Adverb
hope	Yes	No	Yes	No
be after	No	No	Yes	No

As verbs, be after is a hypernym of hope; that is, be after is a word with a broader meaning than hope:

Other hypernyms of hope include plan.

hope (noun)	be after (noun)
one of the three Christian virtues
grounds for feeling hopeful about the future
the general feeling that some desire will be fulfilled
a specific instance of feeling hopeful
someone (or something) on which expectations are centered

hope (verb)	be after (verb)
intend with some possibility of fulfilment	have the will and intention to carry out some action
be optimistic; be full of hope; have hopes
expect and wish