Without objection

Solution 1:

Complying without objecting would be described by acquiescence (passive acceptance):

His acquiescence to her strict instructions

Although technically, that would mean agreeing to them, not necessarily complying with them.

https://www.merriam-webster.com/dictionary/acquiescence

Solution 2:

Willing - Having a ready will; disposed to consent or comply; ready to do (what is specified or implied) without reluctance, having no objection, ‘not disposed to refuse’ (Johnson); spec. disposed to do what is required, ready to be of use or service.(OED)

His willing compliance with her strict instructions.

Solution 3:

I think that you are looking for unquestioning.

His unquestioning compliance with her strict instructions

https://www.merriam-webster.com/dictionary/unquestioning

Definition of unquestioning

not questioning : not expressing or marked by doubt or hesitation

unquestioning obedience; unquestioning loyalty

Solution 4:

You might also consider tacit, defined by Merriam Webster as:

: implied or indicated (as by an act or by silence) but not actually expressed <tacit consent> <tacit admission of guilt>