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JavaScript

var obj1 = {
    "AMLookupTasksList": [{
        "Id": 1,
            "Title": "Create and Maintain an Onboarding Document",
            "Responsibility": "1. Onboarding|f101ccf1-c7d5-42e7-ba8f-48e88ac90a3d"
    }, {
        "Id": 2,
            "Title": "Execute Onboarding for New Consultants",
            "Responsibility": "1. Onboarding|f101ccf1-c7d5-42e7-ba8f-48e88ac90a3d"
    }]
};
var obj2 = {
    "AMTasksList": [{
        "Id": 4,
            "Title": {
            "$M_1": 13,
                "$c_1": "Create and Maintain an Onboarding Document"
        },
            "Responsibility": null
    }, {
        "Id": 17,
            "Title": {
            "$M_1": 12,
                "$c_1": "Execute Onboarding for New Consultants"
        },
            "Responsibility": null
    }]
};

var lookup = {};
// create refernece to list above and use it everywhere
lookup.list = obj1;
for (var i = 0, len = lookup.list.AMLookupTasksList.length; i < len; i++) {
    lookup[lookup.list.AMLookupTasksList[i].Title] = lookup.list.AMLookupTasksList[i];
}
var otherLookup = {};
otherLookup.list = obj2;
for (var i = 0, len = otherLookup.list.AMTasksList.length; i < len; i++) {
    otherLookup[otherLookup.list.AMTasksList[i].Title.$c_1] = otherLookup.list.AMTasksList[i];
}

// copy value for Responsibility from first to second
var i = 0;
var len = otherLookup.list.AMTasksList.length;
for (i; i < len; i++) {
    if (lookup[otherLookup.list.AMTasksList[i].Title.$c_1]) {

        otherLookup.list.AMTasksList[i].Responsibility = lookup[otherLookup.list.AMTasksList[i].Title.$c_1].Responsibility;
    }
}

alert(otherLookup["Create and Maintain an Onboarding Document"].Responsibility);